19730012271_1973012271 nasa tn d-7228

NASA TECHNICAL NOTE NASA TN 0-7228

00 cv N

CZI 9

A METHOD FOR ESTIMATING STATIC AERODYNAMIC CHARACTERISTICS FOR SLENDER BODIES OF CIRCULAR AND I

NONCIRCULAR CROSS SECTION ALONE A N D WITH LIFTING SURFACES AT ANGLES OF ATTACK FROM Oo TO 90"

by Lelund H. Jorgensen

Ames Reseurch Center Moffett Field, CuliJ: 94035

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, 0. C. * APRIL 1973

Leland H. Jorgensen

501 -06-05-00-21

Technical Note National Aeronautics and Space Administration Washington, D. C., 20546

I 15. Supplementary Notes

16. Abstract

An engineering-type method is presented for estimating normal-force, axial-force, and pitching-moment coefficients for slender bodies of circular and noncircular cross section alone and with lifting surfaces. Static aerodynamic characteristics computed by the method are shown to agree closely with experimental results for slender bodies of circular and elliptic cross section and for winged-circular and wingedelliptic cones. However, the present experimental results used for comparison with the methott are limited to angles of attack only up to about 20" and Mach numbers from 2 to 4.

17. Key Words (Suggested by Author(s))

Body theory Noncircular body theory Wing-body theory High angles of attack

19. Security Classif. (of this report) I 20. Security Classif.

18. Distribution Statement

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I I $3.00 -

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* For sale by the National Technical Information Service, Springfield, Virginia 22151

TABLE OF CONTENTS

Page

NOTATION. V

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

PROCEDURE AND FORMULAS FOR COMPUTING AERODYNAMIC CHARACTERISTICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 . 5 6 '

BodyAloneMethodofReference9- . . . . . . . . . . . . . . . . . . . . . General Case for aBody Alone orWith Lifting Surfaces . . . . . . . . . . . . . Special Case for an Elliptic Cone With Triangular Wing Formulas and Values of (cn/cno)sB and (Cn/Cno)Newt for Winged-Elliptic

. . . . . . . . . . . . Cross Sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

.% (Cn/Cno)sB formulas . . . . . . . . . . . . . . . . . . . . . . . . 7 (Cn/Cno)Newt formulas . . . . . . . . . . . . . . . . . . . . . . . . 9 Values of (cn/cno)sB and (Cn/Cno)Newt . . . . . . . . . . . . . . . . . 9

COMPARISON OF COMPUTED WITH EXPERIMENTAL AERODYNAMIC CHARACTERISTICS. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bodies with circular and elliptic cross sections . . . . . . . . . . . . . . . Winged-circular and winged-elliptic cones . . . . . . . . . . . . . . . . .

Computation of Aerodynamic Characteristics. . . . . . . . . . . . . . . . . .

Circular and Elliptic Cross Sections . . . . . . . . . . . . . . . . . . . . .

Models Studied and Test Conditions . . . . . . . . . . . . . . . . . . . . .

Comparison of Computed With Experimental Characteristics for Bodies With

Comparison of Computed-With Experimental Characteristics for Winged-Circular &d Winged-Elliptic Cones. . . . . . . . . . . . . . . . . . . . . . . . .

10 10 10 10 11

11

12

. . . . . . . . . . . . . . . . . . 13 CONCLUDING REMARKS - 8 . - - * - * *

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

iii

A

NOTATION

body cross-sectional area

Ab

AP

Ar

a, b

CL

d

Fa, Fn

(I

body base area (at x = (I)

planform area

reference area (taken as A b for the comparisons of computed with experimental results)

semimajor and semiminor axes of elliptic cross section

Fa axial-force coefficient, -

q J r crossflow drag coefficient of circular cylinder section.

drag drag coefficient, -

4 - 4

>

Fn

qn(AQcy)dcy

lift lift coefficient, - q,Ar

pitching moment

4,Ai-X pitching-moment coefficient about station at Xm from nose,

hn normal-force coefficient, -

q,Ar

local normal-force coefficient per unit length P - P ,

4 , pressure coefficient, -

body cross-section diameter

axial and normal forces

body length

Mach number component normal to body axis, M , sin a

free-stream Mach number

pressure

freestream static pressure

dynamic pressure component normal to body axis, q sin' a

free-stream dynamic pressure, - p V,'

00

1 2

V

r

S

Re

Ren

V

Vn

vw W

X

X

Xac

X C

X m

(x

P

e

rl

7

P

P

cp

body cross-section radius

semispan

free-stream Reynolds number, - d

Reynolds number component normal to body axis, Re - sin CY X body volume

velocity component normal to body axis, Vw sin a

P vwx P

free-stream velocity

body width

reference length

axial distance from body nose

distance from nose to aerodynamic force center

distance from nose to centroid of body planform area

distance from nose to pitching-moment reference center

angle of attack

wing planform semiapex angle

crossflow drag proportionality factor

ratio of the lift of a triangular wing by linearized theory to the lift by slender-body theory

viscosity coefficient of air

density of air

angle of bank about body longitudinal axis

Sub scripts

C Y cylinder

Newt Newtonian theory

vi

0

SB slender-body theory

stag stagnation

equivalent circular body or cross section

Vii

A METHOD FOR ESTIMATING STATIC AERODYNAMIC CHARACTERISTICS FOR SLENDER BODIES OF CIRCULAR AND NONCIRCULAR CROSS SECTION

ALONE AND WITH LIFTING SURFACES AT ANGLES OF ATTACK FROM 0' TO 90"

Leland H. Jorgensen

Ames Research Center

SUMMARY

An engineering-type method is presented for estimating normal-force, axial-force, and pitching- moment coefficients for slender bodies of circular and noncircular cross section^ alone and with lifting surfaces. In the generalized equations that are given for CN and Cm, ratios are required of the local normal-force coefficient per unit length for the cross section of interest to that for the equivalent circular cross section. These ratios are given both from slender-body and Newtonian theories. Formulas and numerical values of these ratios for winged-elliptic cross sections &e included in the report.

Static aerodynamic characteristics computed by the method are shown to agree closely with experimental results for slender bodies of circular and elliptic cross section and for winged-circular and winged-elliptic cones. However, because the experimental results are limited to angles of attack of less than about 20' and Mach.numbers only from 2 to 4, further comparison of the method with more data is needed to determine validity limits for the method. The method may be applicable or adaptable for use at subsonic, supersonic, and low hypersonic Mach numbers.

INTRODUCTION

High angle-of-attack aerodynamics is increasing in importance because of the demand for greater maneuverability of space shuttle vehicles, missiles, and military aircraft (both manned and remotely piloted). At present there appears to be a lack of analytical methods and aerodynamic data applicable to the design of advanced configurations for flight at high angles of attack over a wide range of Mach and Reynolds numbers.

The purpose of this report is to present an engineering-type method for estimating the normal- force, axial-force, and pitching-moment coefficients for slender bodies of circular and noncircular cross section alone and with lifting surfaces (such as wings and tails) at angles of attack from 0" to 90". Effects of body and wing vortex flows on tail surfaces and controls are not included, and a complete analysis of wing-body-tail aerodynamics and control effectiveness is not pursued.

The method may be applicable or adaptable for use at subsonic, supersonic, and low hypersonic Mach numbers, although its true limits of applicability must await experimental verification. In the present report, computed force and moment characteristics are compared with available experimental

results for bodies of circular and elliptic cross section and for winged-circular and winged-elliptic cones at angles of attack from 0" to about 20' and Mach numbers from about 2 to 4.

PROCEDURE AND FORMULAS FOR COMPUTING AERODYNAMIC CHARACTERISTICS

Prior to the work of Allen in 1949-5 1 (refs. 1 and 2) most analytical procedures for computing the aerodynamic characteristics of bodies and wing-body combinations were based on potential theory and were limited in usefulness to very low angles of attack. Allen proposed a method for predicting the static longitudinal forces and moments for bodies of revolution inclined to angles of attack considerably higher than those for which theories based only on potential-flow concepts are known to apply. In this method a crossflow lift attributed to flow separation is added to the lift predicted by potential theory. This method has been used quite successfully in computing the aerodynamic coefficients of inclined bodies (e.g., refs. 1-6), although most data available for study until 1961 were for bodies at angles of attack below about 20°, and the formulas were initially written to apply only over about this angle-of-attack range.

In 196 1 , Allen's concept (ref. 7) was adopted for computing the normal-force, axial-force, and pitching-moment coefficients for a rocket booster throughout the angle of attack range from Oo to 180". Satisfactory agreement of theory with experiment was obtained for a test model of the rocket booster over the Mach number range from 0.6 to 4. Further application of the Allen concept was made by Saffell, Howard, and Brooks (ref. 8) in 1971 in a programmed method for predicting the static longitudinal aerodynamic characteristics of low aspect-ratio missiles operating at angles of attack up to 180".

In 1958, a method for computing the aerodynamic characteristics for bodies of noncircular cross section was proposed (ref. 6). In this method, normal-force and pitching-moment coefficients ( C N ~ and Cmo) are computed by Allen's formulas for the equivalent body of revolution which has the same axial distribution of cross-sectional area as the noncircular body. Then the values of CN and Cm for the noncircular body are computed from CN/CN and Cm/Cmo ratios determined from apparent mass coefficients (i.e., from slender-body theory). 8ood agreement of theory with experiment (ref. 6) was obtained by this procedure for bodies of elliptic cross section at the conditions investigated (a/b's from 1 to 2, cp's of 0" and 90", Moo's from 2 to 4, and a's from 0" to 20").

Recently, in 1972, the Allen concept again has been applied in the development of an engineering- type procedure (ref. 9) for computing normal-force, axial-force, and pitching-moment coefficients for slender bodies of circular and noncircular cross sections at angle of attacks from 0" to 180". The CN and Cm formulas are written, however, for a body whose cross-sectional shape remains constant over the body length, but the cross-sectional area, of course, is allowed to vary.

In this section of the present report, the method of reference 9 is first reviewed. Then CN and Cm expressions are written for the general case of a body alone or with lifting surfaces (e.g., wings) where the cross-sectional shape, as well as the cross-sectional area, is allowed to vary along the body length. For the special case of winged-elliptic cones, simplified expressions for C' and Cm are also presented.

2

In all of these expressions for CN and Cm, it is necessary to have values of local normal-force coefficient per unit length for the noncircular cross sections (Cn) ratioed to those for the equivalent circular body cross sections (Cao). Formulas for computing C&,, ratios for winged-elliptic cross sections are given, and computed values for some typical cases are presented.

Body Alone Method of Reference 9

For a slender body whose cross-sectional shape is constant over its length, Jorgensen (ref. 9) has suggested equations for computing the normal-force, axial-force, and pitching-moment coefficients. For the sign convention in sketch (a) and for a's from 0" to 90"' these equations are

A b c N = - sin 2a cos 5 Ar (2) cNo sg

+qCd A P - sin' a (1) Ar

CA = CAa = ooCOS2 (Y (2) and Sketch (a)

The aerodynamic force center is then given by

where X is the reference length.

The first terms in equations (1) and (3) come from slender-body potential theory. The second terms represent the viscous crossflow or crossflow attributed to flow separation.

In equation (l), ( C N / C N ~ ) is the ratio of the normal-force coefficient for the body of noncircular cross section to that for the equivalent body of circular cross section (i.e., the circular body hav- ing the same area distribution). The ratio ( C N / C N , ) ~ ~ is determined from slender-body theory, and the ratio ( C ~ , T / C ~ ~ , T ~ ) ~ ~ ~ ~ is determined from Newtonian impact theory. For a body whose cross-

3

sectional shape is constant over its length, these ratios are equal to the ratios of the normal-force coefficients per unit length; that is,

and

Newt O Newt O Newt

In equations (1) and (3), Cdn is the crossflow drag coefficient for a section of an “infinite length” or truly two-dimensional circular cylinder placed normal to an airstream. It is a function of both the Mach number and Reynolds number components that are normal to the cylinder longitudinal axis, and hence for a body at angle of attack it is a function of

and MU =Moo sin a

Ren = R e sin a

M n is commonly called the crossflow Mach number and Ren the crossflow Reynolds number. Necessary “state-of-the-knowledge,’ plots of cd , versus M n and Ren are given in reference 9.

The crossflow drag proportionality factor is q, that is, the ratio of the crossflow drag coefficient for a finite length cylinder to that for an infinite length cylinder. In reference 9, q’s from reference 10 are plotted as a function of length-to-width ratio for both circular cylinders and flat plates. It is suggested in reference 9 that q’s for noncircular as well as for circular bodies be estimated from this plot for bodies at subsonic free-stream Mach numbers. Because the curves for the circular cylinders and flat plates lie close together, estimates for noncircular bbdies are easy to make. For bodies at supersonic and hypersonic free-stream Mach numbers, experience to date has shown that it is best to assume that q = 1 (ref. 9).

In the method of reference 9, C’ and Cm are controlled through Cdn by either crossflow Mach number M n or crossflow Reynolds number Ren. Equations (1) and (3) have been written for the case of CN and Cm controlled by Mn. It might be instructive to emphasize that there is some basic experimental justification, even at low subsonic M n , for the use of Newtonian theory to determine the ratio CN/CN, (= Cn/Cn,) in the second term of equation (1). For various rounded and blunt two- dimensional cylinders (table l), values of Cd and Cn/Cn, computed by Newtonian and modified Newtonian theories agree reasonably well with measured results for subcritical values of M n and Ren. This agreement is shown in table 1 for the circular, elliptical, and square cross-sectional shapes considered.

n

For bodies at low subsonic Mn7s (below critical), the variation of Cdn with Ren for the cross section of interest can become significantly large as Ren exceeds the critical value. For this Ren controlled condition, equations (1) and (3) can be used with slight modification to the second

4

terms. (C'/CjvJNewt and (Cm/Cmo)~ewt are removed, and experimental values of Cdn are used for the cross section of interest. Note, however, that most experimental values of Cdn are based on cross-sectional width w and must be multiplied by w/d, where d is the equivalent diameter of the cross section. In reference 9 a compilation of references is given from which experimental values of Cdn vs Re, can be obtained for various designated cross sections and flow directions.

The reader is referred to reference 9 for review of equations and methods for computing wave, skin-friction, and base-pressure contributions to CA and for a method other than equation (2) for computing CA .

TABLE 1.- cdn AND C,C' VALUES FOR TWO-DIMENSIONAL CYLINDERS OF VARIOUS CROSS SECTIONS AT a, = 90" AS COMPUTED BY NEWTONIAN THEORIES AND MEASURED AT SUBCRITICAL MACH AND REYNOLDS NUMBERS

CROSS SECTION

k = 0.0 , f~?j k = 0.02 k = 0.08 k = 0.24 k= 0.50

MOD. NEWT. THEORY NEWTONIAN THEORY FOR c =1.8 MEASURED

Pstag

I I I I 1.33 I 1.00 I 1.20 I 1.00 I 1.20 I 1.00 I I I

I I I I I I I I I I I

I I

~

1.68 I 1.14

0.50 0.22 0.35 I 0.15 I 12

1.75 I .60 I 1.89 I 13,14

0.70 I 0.41 I 12, 13

I I

I I

I I

~

1.14 1.12 I 0.85 I 15 1.33 i 1.00 I 1.20 I 1.00 I 1.20 i 1.00 i II

NOTE: ALL Cd,'s IN TABLE ARE BASED ON WIDTH OF CROSS SECTION, NOT EQUIVALENT d.

General Case for a Body Alone or With Lifting Surfaces

For the more general case of a body alone or with lifting surfaces where the cross-sectional shape can vary along the body length, a procedure somewhat similar to that of reference 9 is suggested. However, for the general case, ratios of (Cn/Cn,)sB and (Cn/Cno)Newt at axial stations along the body length must be used, and the crossflow terms predicted from slender-body potential theory and from viscous crossflow theory must be written in integral form. For positive dA/dx values,

5

and

In equations (7) and (8) the first terms (from slender-body theory) are not applicable, as written, for winged-body sections where the body dA/dx values are zero or negative. For such uses, procedures similar to those suggested in reference 16 probably should be employed to account for the potential- flow contribution of the wing to CN and Cm. Also, for many wing-body-tail configurations, effects of vortices from body and forward lifting surfaces should be considered, and a vortex interference procedure similar to that of reference 16 probably should be formulated in conjunction with the present method.

For some applications it might be preferable to write CN in a form similar to equation (1). When this is done,

(9) R a

where the terms (l/Q)$ (Cn/Cno)s~ dx and (l/Q)$ (Cn/Cno)Newt dx, represent average values

of (Cn/Cno )SB and (CG/Cno)Newt over the body length. 0 0

Special Case for an Elliptic Cone With Triangular Wing

For a cone of elliptic cross section with a triangular (delta) wing of the same length as the cone (sketch (b)), equations (7) and (8) for CN and Cm simplify to equations (1) and (3) , but values of ( C N / C N ~ ) = (Cm/Cmo) for the wing-body cross section must be used. In addition, further refinement to the slender-body term appears warranted.

Reference 17 shows that slender-body theory for triangular-winged bodies can be modified to give results comparable to linearized theory. This is accomplished merely by multiplying the slender- body result by a modification factor A. This factor is the ratio of the lift of the wing alone by linearized theory to that by slender-body theory and is given by

I for 0 tan E < 1 (subsonic leading edge) 1 A =

~(d1 - p 2 tan2 E

and (10)

A = for 0 tan E 2 1 (supersonic leading edge) tan E

Here E( ) is a complete elliptic integral of the second kind. 6

and

Sketch (b)

Equations (1) and (3) thus can be modified to give

(1 1)

where

and

Formulas for computing values of (Cn/Cno)sB and (Cn/Cno)Newt for winged-elliptic cross sections are presented next. They are not restricted in their use, however, only to winged-elliptic cones.

Formulas and Values of (cn/cno)sB and (Cn/Cno)Newt for Winged-Elliptic Cross Sections

fCn/Cno)sB fomzuZas.- From slender-body theory (e.g., refs. 18-21) the ratio of Cn for a winged-body cross section to that for the equivalent (same area) circular-body cross section can be

7

determined for many cross-sectional shapes. In the present study (Cn/Cn,)s~ expressions have been determined for winged-circular and winged-elliptic cross sections (see sketches (c), (d), and (e)).

L

Sketch (c) Sketch (d) Sketch (e)

For a winged-circular cross section with the wing planform perpendicular to the crossflow velocity Vn (sketch (c)),

For a winged-elliptic cross section with the semimajor axis a and wing planform perpendicular to the crossflow velocity V , (sketch (d)),

where

and 1 2

ul = -(s +.Js2 + b2 -a2)

For a winged-elliptic cross section with the semiminor axis b and wing planform perpendicular to the crossflow velocity V, (sketch (e)),

8

where (a + b), k, = (1, - ___

402

and

1 2 02 = - ( s +ds2 +a2 - b2>

(Cn/Cno,J~ewt formulas. - From Newtonian impact theory, (Cn/Cno)~ewt expressions also have been derived for winged-circular and winged-elliptic cross sections.

For a winged-circular cross section with the wing planform perpendicular to the crossflow velocity Vn (sketch (c)),

For a winged-elliptic cross section with the semimajor axis a and wing planform perpendicular to the crossflow velocity Vn (sketch (d)),

log [t (I + dG)]+-+ 1 s -- 1 I (17) b2 a 1 - - U 2

For a winged-elliptic cross section with the semiminor axis b and wing planform perpendicular to the crossflow velocity Vn (sketch (e)),

Vulues of (Cn/Cno)s~ and (Cn/Cno)~ewp- From equations (1 3) through (1 8), values of (Cn/Cno)s~ and (Cn/Cno)~ewt have been computed for elliptic cross sections alone and with wings. The results are plotted and compared in figures 1 through 4.

In figure 1 the variation of (Cn/Cno) with axis ratio a/b is given for an elliptic cross section without wings. As previously noted in reference 9, values of (Cn/Cno) from slender-body theory are reasonably close to those from Newtonian theory for many a/b's of interest.

In figure 2 the variation of (C&'no) with the ratio of wing semispan s to body radius Y is given for a winged-circular cross section. For s/r's less than about 2, the values from both theories are reasonably close to each other, but with further increase in s/r the values of (Cn/Cno)s~ greatly exceed those of (Cn/Cno)~ewt.

9

In figure 3 values of (Cn/Cmo) are presented for a winged-elliptic cross section with the semimajor axis a perpendicular to the crossflow velocity Vn. For the axis ratios of a/b = 2 and 3, the figure gives the variation of (Cn/Cno) with the ratio of semispan s to semimajor axis a. As either a/b or s/a increases, the disagreement between the results from the theories increases.

In figure 4 values of (Cn/Cno) are presented for a winged-elliptic cross section with the semiminor axis b perpendicular to the crossflow velocity VU. For axis ratios of a/b = 2 and 3, the variation of (Cn/Cno) with s/b is given. There is closer agreement between the values computed from the two theories for this cross section arrangement than for the arrangement where the semimajor axis and wing are perpendicular to Vn.

COMPARISON OF COMPUTED WITH EXPERIMENTAL

AERODYNAMIC CHARACTERISTICS

In the present study, computed longitudinal aerodynamic force and moment characteristics have been compared with experimental results (ref. 6) for bodies with circular, and elliptic cross sections and with experimental results (ref. 22) for winged-circular and winged-elliptic cones.

Models Studied and Test Conditions

Bodies with circular and elliptic cross sections.- Drawings of the studied bodies with circular and elliptic cross sections are shown in figure 5. The bodies' overall fineness ratios (Qld's) are 6 and 10, and they all have fineness-ratio-3 noses followed by cylindrical sections. The bodies of elliptic cross section (a/b = 2) have the same cross-sectional area distribution as the equivalent circular bodies, and they were tested both oriented at cp = 0" and cp = 90", as shown in figure 5.

Lift, drag, and pitching-moment coefficients were measured for these bodies in the NASA Ames 1- by %Foot Supersonic Wind Tunnel No. 1. All bodies were tested at a free-stream Mach number

M , of 1.98. Only the fineness-ratio-10 bodies were tested at M , = 3.88. The Reynolds number, based on body length, was 6.7X lo6 for the fineness-ratio-10 bodies and 4.OX1O6 for the fineness- ratio-6 bodies. The angle-of-attack range was from 0" to about 22" for the bodies at M , = 1.98 and from 0" to about 15" for the bodies at M , = 3.88. All drag coefficients presented in reference 6 have had the effects of base pressure removed and do not include base pressure drag.

Winged-circular and winged-ellip tic cones. - Drawings of the winged-circular and winged-elliptic cones tested in reference 22 are shown in figure 6. Triangular wings of aspect ratio 1 .O and 1.5 were tested in combination with circular cones and elliptic cones, all of fineness-ratio-3.67. As shown in figure 6, the elliptic cones of a/b = 3 were arranged both with the semimajor axis a in the wing plane and perpendicular to it.

Lift, drag, and pitching-moment coefficients were measured for these models at M,'s of 1.97 and 2.94 in the NASA Ames 1- by %Foot Supersonic Wind Tunnel No. 2 (a tunnel since disassem- bled). The Reynolds number, based on model length, was about 8.0X lo6, and the angles of attack ranged from 0" to about 16'. All drag coefficients presented in reference 22 have had the effects of body base pressure removed and do not include base pressure drag. 10

Computation of Aerodynamic Characteristics

Because CL, CD, LID, and xac/R data are given in references 6 and 22, these terms were computed for the models and test conditions considered. Since CN and CA values are computed from the formulas of this report, it was necessary to compute CL and CD values with the transformation expressions :

In the computation of C' from equation (2), values of C' were assumed to be the same as

those computed in references 6 and 22. The reader who is interested in computing values of CA W O O

for ogival and conical nosed bodies of revolution and for elliptic cones is also referred to the procedures and formulas cited in reference 9.

F O "

Equations (1) and (3) were used to compute CN and Cm values for the bodies with circular and elliptic cross sections. Equations (1 1) and (12) were used to compute these values for the winged- circular and winged-elliptic cones.

Comparison of Computed With Experimental Characteristics for Bodies With Circular and Elliptic

Cross Sections

In figures 7 through 9, computed values of C c Co, LID, and xuc/R as a function of angle of attack a are compared With the experimental results for the bodies of R/d = 6 and 10 at M , = 1.98 and R/d = 10 at M , = 3.88. Generally, there is very good agreement of the computed with the experimental results. It is encouraging that effects of cross section (u/b), fineness ratio (Q/d), and Mach number (M-) on all of the aerodynamic characteristics are predicted so well.

Because of lack of data for a's greater than about 20" , there is uncertainty concerning the validity of the method for use throughout the a range from 0" to 90". However, because of the close agtee- ment of computed with experimental results, shown in reference 9, for a series of cylinder, cone- cylinder, and ogive-cylinder bodies of revolution at a's from 0" to 180" with M , = 2.86, it is strongly believed that the method also will correctly predict the characteristics for the elliptic bodies throughout the a range. Nevertheless, over the M,, Re, and a ranges of current interest, further testing of bodies with elliptic and other cross sections appears desirable to ascertain validity limits for the method.

11

Comparison of Computed With Experimental Characteristics for Winged-Circular

- I I I 1

and Winged-Elliptic Cones

In figures 10 through 13, computed aerodynamic characteristics are compared with the experimental results for the winged-circular and winged-elliptic cones. As for the bodies alone, the computed and experimental results generally agree. The closest agreement is at M , = 1.97 (see figs. 10 and 11). At M , = 2.94 (figs. 12 and 13) themethod tends to overpredict CL somewhat with increase in a. However, the agreement is still probably acceptable for most engineering studies.

In the method for computing the aerodynamic characteristics, values of crossflow drag coefficient cd for a two-dimensional circular cylinder are used. Generally cdn is a function only of M n (Mn = M , sin a),except for Re greater than about 2X lo5 with M n less than about 0.5 (see, e.g., ref. 9). With increase in M n from 0.2 to 1, Cdn increases from about 1.2 to 2, then decreases asM, increases into the supersonic-hypersonic regime (see sketch (f)). At hypersonic M u , values of Cdn predicted from modified Newtonian theory agree closely with experiment. From modified Newtonian theory,

n

= 1.2 for C = 1.8 Pstag

For M n greater than about 4, C 1.8 from perfect-gas relations. Pstag

r n ‘Modified Newtonian theory [ \

t

Sketch (f)

12

It is questionable whether the experimental variation of Cdn with M n for a circular cylinder should be used in the calculation of the aerodynamic characteristics for winged bodies. It is likely that some other variation or a constant value of Cdn will give closer agreement of theory with experiment, especially at a's where M n is in the transonic regime.

Figure 14 indicates the effect of Cdn on the prediction of CL and LID for the winged cones at a's from 0" to 90" for M , = 2.94. Computed curves are shown for the assumptions that Cdn = fi'Mn) and Cdn = 1.2, the value for both low subsonic and hypersonic M n for a circular cylinder. The figure shows significant effect of Cdn on the prediction of CL for all models at a's greater than about 15" for M , = 2.94. There is negligible effect, however, onthe prediction of LID. Further testing of winged bodies at high angles of attack is definitely necessary to aid in the development of correct usage of the present method for computing CL.

CONCLUDING REMARKS

An engineering-type method has been presented for estimating normal-force, axial-force, and pitching-moment coefficients for slender bodies of circular and noncircular cross section alone and with lifting surfaces, In the generalized equations that are given for C~\J and C&, ratios are required of the local normal-force coefficient per unit length for the cross section of interest to that for the equivalent (same area) circular cross section. These ratios are given both from slender-body and New- tonian theories. Formulas and numerical values of these ratios for winged-elliptic cross sections are included in this report.

Static aerodynamic characteristics computed by the method have been shown to agree closely with experimental results for slender bodies of circular and elliptic cross section and for winged- circular and winged-elliptic cones. However, because the experimental results are limited to angles of attack less than about 20" and Mach numbers only from 2 to 4, further comparison of the method with more data is needed to determine validity limits for the method.

Effects of forebody-flow and wing-flow separation on downstream aerodynamic surfaces have not been included in the present study, but these effects should be investigated by representing regions of separated flow with both concentrated and distributed vortices. Then the effects of inte- grated forces and moments caused by varying configuration geometry can be studied numerically. However, confidence in the analytical approaches will have to be established by comparison of computed with experimental results for wing-body-tail combinations at high angles of attack (up to 90") and at Mach and Reynolds numbers of interest.

Ames Research Center National Aeronautics and Space Administration

Moffett Field, Calif., Dee. 13, 1972

13

REFERENCES

1. Allen, H. Julian: Estimation of the Forces and Moments Acting on Inclined Bodies of Revolution of High Fineness Ratio. NACA RM A9126,1949.

2. Allen, H. Julian; and Perkins, Edward W.: A Study of Effects of Viscosity on Flow Over Slender Inclined Bodies of Revolution. NACA Rep. 1048, 1951.

3. Perkins, Edward W.; and Kuehn, Donald M.: Comparison of the Experimental and Theoretical Distributions of Lift on a Slender Inclined Body of Revolution at M = 2. NACA TN 3715,1956.

4. Perkins, Edward W.; and Jorgensen, Leland, H.: Comparison of Experimental and Theoretical Normal-Force Distributions (Including Reynolds Number Effects) on an Ogive-Cylinder Body at Mach Number 1.98. NACA TN 3716,1956.

5. Jorgensen, Leland H.; and Perkins, Edward W.: Investigation of Some Wake Vortex Characteristics of an Inclined OgiveXylinder Body at MachNumber 2. NACA Rep. 1371,1958.

6. Jorgensen, Leland H.: Inclined Bodies of Various Cross Sections at Supersonic Speeds. NASA MEMO 10-3-58AY 1958.

7. Jorgensen, Leland H.; and Treon, Stuart L.: Measured and Estimated Aerodynamic Characteristics for a Model of a Rocket Booster at Mach Numbers From 0.6 to 4 and at Angles of Attack From 0" to 180". NASA TM X-580, 1961.

8. Saffel, Bernard F., Jr.; Howard, Millard L.; and Brooks, Eugene N., Jr.: Method for Predicting the Static Aero- dynamic Characteristics of Typical Missile Configurations for Angles of Attack to 180 Degrees. Report 3645, Naval Ship Research and Development Center, March 1971.

9. Jorgensen, Leland H. : Prediction of Static Aerodynamic Characteristics for Space-Shuttle-Like and Other Bodies at Angles of Attack From 0" to 180". NASA TN D-6996,1973.

10. Goldstein, Sydney: Modern Developments in Fluid Dynamics. Oxford, The Claredon Press, vol. 2, sec. 195, 1938, pp. 439-440.

11. Wieselsberger, C.: New Data on the Laws of Fluid Resistance. NACA TN 84, 1922.

12. Lindsey, W. F.: Drag of Cylinders of Simple Shapes. NACA Rep. 619, 1938.

13. Delany, Noel K.; and Sorensen, Norman E.: Low-Speed Drag of Cylinders of Various Shapes. NACA TN 3038, 1953.

14. Polhamus, Edward C.; Getler, Edward W.; and Grunwald, Kalman J.: Pressure and Force Characteristics of Non- circular Cylinders as Affected by Reynolds Number With a Method Included for Determining the Potential Flow About Arbitrary Shapes. NASA TR R-46,1959.

15. Polhamus, Edward C.: Effect of Flow Incidence and Reynolds Number on Low-Speed Aerodynamic Charac- teristics of Several Noncircular Cylinders With Applications to Directional Stability and Spinning. NASA TR R-29, 1959.

14

16. Pitts, William C.; Nielsen, Jack N.; and Kaattari, George E.: Lift and Center of Pressure of Wing-Body-Tail Combinations at Subsonic, Transonic, and Supersonic Speeds. NACA Rep. 1307,1957.

17. Nielsen, Jack N.; Katzen, Elliott D.; and Tang, Kenneth IC: Lift and PitchinRMoment Interference Between a Pointedcylindrical Body andTriangular Wings of Various Aspect Ratios at Mach Numbers of 1.50 and 2.02: NACA TN 3795,1956.

18. Bryson, Arthur E., Jr.: Stability Derivatives for a Slender Missile With Application to a Wing-Body-Vertical-Tail Configuration. Jour. Aero. Sci., vol. 20, no. 5, May 1953, pp. 297-308.

19. Bryson, Arthur E., Jr.: Evaluation of the InertiaCoefficients of thecross Section of a Slender Body. Jour. Aero. Sci., vol. 21, no. 6, June 1954, pp. 424-427.

20. Bryson, Arthur E., Jr.: The Aerodynamic Forces on a Slender Low (or Hi&) Wing, Circular Body, Vertical Tail Configuration. Jour. Aero. Sci., vol. 21, no. 8, Aug. 1954, pp. 574-575. ,

21. Nielsen, Jack N.: Missile Aerodynamics. New York, McGraw-Hill Company, Inc., 1960.

22. Jorgensen, Leland H.: Elliptic Cones Alone and With Wings at Supersonic Speeds. NACA TN 40:45, 1957.

15

" I 2 3 4 5 6

a /b

(a) Semimajor axis a perpendicular to crossflow velocity V,.

I 2 3 4 5 6 a/b

(b) Semiminor axis b perpendicular to crossflow velocity V,.

Figure 1 .- Ratio of local normal-force coefficient for an elliptic cross section to that for the equivalent circular cross section.

A-4700

s /r

Figure 2.- Ratio of local normal-force coefficient for a winged-circular cross section to that for the equivalent circular cross section.

18

(a) a/b = 2

(b) a/b = 3

Figure 3.- Ratio of local normal-force coefficient for a winged-elliptic cross section to that for the equivalent circular cross section; semimajor axis a perpendicular to crossflow velocity Vrz.

s/b

(a) a/b = 2

s/b

(b) a/b = 3

Figure 4.- Ratio of local normal-force coefficient for a winged-elliptic cross section to that for the equivalent circular cross section; semiminor axis b perpendicular to crossflow velocity Vn -

20

A 0 td=3.56 cm

a =.707d

-=2 a b +=goo

Td = 3.56 cm

-7- 0 a = .707d

-=2 a b

Figure 5.- Models for which the aerodynamic characteristics were measured in reference 6 and computed in the present study.

21

.005 d radius / / / / /

Typical leading edge Section A-A (Enlarged)

7 cm

a/b=l s/a = 1.84

4 1

a/b=3 s/a=1.06

a/b=3 s/b=3.18 9';

(a) Wing aspect ratio = 1.0.

~ ~ ~ 0 0 5 d radius

Typical trailing edge Section B-B (Enlarged)

s/a=2.76

a /b=3 s/a = 1.59

a/b=3 s/b= 4.77

(b) Wing aspect ratio = 1.5.

Figure 6.- Models for which the aerodynamic characteristics were measured in reference 22 and computed in the present study.

A-4100 22

8

6

CL 4

2

0

(a) Lift.

8

6

CD 4

2

0

(b) Drag.

- Computed 0 0 0 Experiment (ref. 6)

0

.8

0 20 40 60 80 a, deg

(c) Lift-drag ratio. (d) Aerodynamic center.

Figure 7.- Comparison of computed with experimental aerodynamic characteristics for bodies with elliptic cross sections; LID = 6 , Moo = 1.98, Re = 4.0X lo6.

A-4700 23

16

12

CL

4

0

4.c

3.2

2.4

4 D

I .6

.8

C

(a) Lift. (b) Drag.

20 40 60 80 a, deg

(c) Lift-drag ratio.

- Computed 0 0 0 Experiment (ref. 6)

0

.4 ac - 1

.8

0 , 20 40 60 80 a, deg

(d) Aerodynamic center.

Figure 8.- Comparison of computed with experimental aerodynamic characteristics for bodies with elliptic cross sections; LID = 1 O,Mm = 1.98, Re = 6.7X 1 06 .

24

(a) Lift.

4.0

3.2

2.4 - D

I .6

.8

0 0 20 40 60 80

a, deg

- Computed 0 0 0 Experiment (ref 6)


Figure 9.- Comparison of computed with experimental aerodynamic characteristics for bodies with elliptic cross sections; LID = 1O,M, = 3.88, Re = 6.7X lo6.

25

CL

I I

2.0

1.6

1.2

.8

.4

0

I (c)

(a) Lift. (b) Lift-drag polar.

Computed Experiment 0 -?-?k/b = 3 " _"

s/a 1.06 ~ *-_

0 -()- $;9;.84 - a/b 3 s i b = 3.18 0

.6

.8 1

0 4 8 12 16 1 .o


(ref 22)

Figure 10.- Comparison of computed with experimental aerodynamic characteristics for elliptic cones with triangular wings of aspect ratio 1 .O at Moo = 1.97; Re = 8X lo6.

26

0 4 8 12 16 Q, deg

(a) Lift.


CD

(b) Lift-drag polar.

Computed t-s? a/b = 3 - s/a=1.59

-0- $::;.76 a /b=3 slb.4.77

.6

.8

I .o 0 4 8 12 16

a, deg


Figure 1 1 .- Comparison of computed with experimental aerodynamic characteristics for elliptic cones with triangular wings of aspect ratio 1.5 at M- = 1.97; Re = 8X 1 06.

A-4700

I .o

Experiment (ref 22) 0

0

0

27

5

4

3 L D -

2

I

0

(a) Lift.

(C)

0 4 8 12 16 Q, d e g



Computed Experiment (ref 22) 0 --I *a/b = 3

s/a = 1.06

-0- $: 11.84 - 0

a/b = 3 s/b = 3.18 0

.6

- 'ac 'I .8

0 4 8 12 16 1.0

a, deg


Figure 12.- Comparison of computed with experimental aerodynamic characteristics for elliptic cones with triangular wings of aspect ratio 1 .O at Moo = 2.94; Re = 8X lo6.

28

-0 4 8 12 16 a, deg

(a) Lift.

(e) Lift-drag ratio.


Computed Experiment k S - l (ref 22) -e] 4 3 + a / b = 3 --- 0 da.1.59 e -0- a/b=l 0 s/a=2.76

0 a/b=3 s/ b = 4.77

.6

xac - 1 .8

I .o 0 4 8 12 16

a, deg


Figure 13.- Comparison of computed with experimental aerodynamic characteristics for elliptic cones with triangular wings of aspect ratio 1.5 at Mco = 2.94; Re = 8X lo6.

8

6

CL 4

2

0 0 IO .20 30 40 50 60 70 80 90

6

5

4

L a 3

2

I

0 0 IO 20 30 40 50 60 70 80 90 A

a, deg

(a) Circular cone.

Figure 14.- Effect of circular-cylinder crossflow drag coefficient Cdn on prediction of CL

and LID for cones with triangular wings at a's from 0" to 90" ; Mm = 2.94.

8

6

Ci 4

2

0

4

L E3

2

I

0

~ ~~~~

0 IO 20 30 40 50 60 70 80 90

0 IO 20 30 . 40 50 60 70 80 90 a deg

(b) Elliptic cone of a/b = 3 with a perpendicular to V,.

Figure 14.- Continued.

8

6

CL

4

2

0 0 I O 20 30 40 50 60 70 80 90

6

5

4

3

2

I

0 0 I O 20 30 40 50 60 70 80 90

a, deg

(c) Elliptic cone of a/b = 3 with b perpendicular to Vn.

32

Figure 14.- Concluded.

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